Dividing a Triangle Into Two Equal Areas

and

Exploring Area Ratios

by

Alison Hays

In this essay, I will explore the line parallel to a base of a triangle that divides the triangle into two equal areas, and I will explore the areas of the sections of the triangle that are created when all three such lines are constructed.

For a statement of this problem, please click here.

There are several parts to this problem, and I will answer each part in turn.

Part 1: Construct a segment parallel to a base of a triangle that divides the triangle into two equal areas.

     For a description of the construction, click here.

     To view a GSP sketch of the construction, click here.

     For a GSP 3 script of the construction, click here.

Part 2: If the parallel segment that divides the triangle into two equal areas is drawn for each base, a smaller triangle is formed. What is the ratio of the area of the small triangle to the original?

Part 3: What is the ratio of the area of the shaded triangle to the area of the original triangle in the figure below? Here again the segments parallel to the bases divide the original triangles into two equal areas.

Part 4: Prove that the measures of the three shaded areas in each of the figures below are the same. In each figure what is the ratio of the area of one of the regions to the area of the original triangle?

Extensions:

Are there any other relationships among areas of the triangle?